Global existence and blow-up for a doubly degenerate parabolic equation system with nonlinear boundary conditions (Q2882495)

The authors study the existence and non-existence of global weak solutions to the following double degenerate parabolic system NEWLINE\[NEWLINE u_{it} = \left(|u_{ix}|^{p_i}(u_i^{m_i})_x\right)_x \qquad (i = 1,2,\dots,k), \qquad x > 0,\qquad 0 < t < T, NEWLINE\]NEWLINE coupled via nonlinear boundary flux, NEWLINE\[NEWLINE -|u_{ix}|^{p_i}(u_i^{m_i})_x(0,t) = \prod_{j=1}^k u_j^{q_i}(0,t) \qquad (i = 1,2,\dots,k), \qquad 0 < t < T, NEWLINE\]NEWLINE with initial data NEWLINE\[NEWLINE u_i(x,0) = u_{i0}(x) \qquad (i = 1,2,\dots,k), \qquad x > 0, NEWLINE\]NEWLINE where \(k \geq 1\), \(m_i \geq 1\), \(p_i > 0\), \(q_{ij} > 0\), and \(u_{i0}\) are nonnegative continuous functions with compact support in \(\mathbb R_+\). It is assumed that the initial data are sufficiently smooth functions and satisfy the compatibility condition. By constructing various kinds of sub- and super-solutions, they give the necessary and sufficient conditions for global existence of non-negative solutions.